| L(s) = 1 | − 2.44·2-s + 5i·3-s − 10·4-s − 12.2i·6-s + (−34.2 − 35i)7-s + 63.6·8-s + 56·9-s + 89·11-s − 50i·12-s − 5i·13-s + (84 + 85.7i)14-s + 4.00·16-s + 485i·17-s − 137.·18-s − 220. i·19-s + ⋯ |
| L(s) = 1 | − 0.612·2-s + 0.555i·3-s − 0.625·4-s − 0.340i·6-s + (−0.699 − 0.714i)7-s + 0.995·8-s + 0.691·9-s + 0.735·11-s − 0.347i·12-s − 0.0295i·13-s + (0.428 + 0.437i)14-s + 0.0156·16-s + 1.67i·17-s − 0.423·18-s − 0.610i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.699 - 0.714i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.699 - 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.5705891101\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5705891101\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + (34.2 + 35i)T \) |
| good | 2 | \( 1 + 2.44T + 16T^{2} \) |
| 3 | \( 1 - 5iT - 81T^{2} \) |
| 11 | \( 1 - 89T + 1.46e4T^{2} \) |
| 13 | \( 1 + 5iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 485iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 220. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + 700.T + 2.79e5T^{2} \) |
| 29 | \( 1 + 191T + 7.07e5T^{2} \) |
| 31 | \( 1 + 1.05e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 1.63e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 2.91e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 377.T + 3.41e6T^{2} \) |
| 47 | \( 1 - 2.19e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 1.58e3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 3.62e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 1.93e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 2.04e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 4.45e3T + 2.54e7T^{2} \) |
| 73 | \( 1 - 8.65e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 5.56e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 1.99e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 808. iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 9.23e3iT - 8.85e7T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.58342079928577546041094216028, −11.01761814261967386079414629129, −10.08045645977499379059266747625, −9.626582182822954813234858643018, −8.512545804993683661880442588519, −7.39747092358553465036904870529, −6.14389559683088763302086749892, −4.40140020118449282209920252349, −3.79298568706156423280299952292, −1.34548999597018740137104132178,
0.29477078972629363047125464032, 1.82717295568717202683512140379, 3.71611305188200093592750498106, 5.17673640080607621292351406789, 6.60300056424660057758041764919, 7.55836283277146200235493668781, 8.757295491932914861369857293896, 9.526667408389423350146632580500, 10.32375671647146119221658163105, 11.93923628532490648680959245430
